I don’t see yet how this connects to the other posts from the epistemology sequence, but it’s definitely nice. I’ve wanted to learn more mathematical logic for some time. I didn’t quite understand why exactly using an axiom schema isn’t as good as using second order logic, before I read this post.
‘There’s no group of numbers G such that for any number x in G, x is the successor of some other number y in G.’
I read that “any” as “for at least one”, rather than as “for every”. That confused me quite a bit. Maybe native speakers won’t have a problem with that, but to me the connection between “any” and “some” is too close.
It’s also not clear to my where the order relation comes from.
I think the point of this post is to demonstrate that logical pinpointing is hard. You might think that the first-order Peano arithmetic axioms logically pinpoint the natural numbers, and what this discussion will end up showing is that they just don’t because of general properties of first-order logic (specifically the Löwenheim–Skolem theorem).
If logically pinpointing something as seemingly simple as the natural numbers depends on something as seemingly nontrivial as understanding the distinction between first-order and second-order logic, then (or so I imagine the argument will continue) we shouldn’t expect logically pinpointing something like morality to be any easier. In fact we have every reason to expect it to be substantially harder.
The definition of the order relation is nontrivial. In second-order Peano arithmetic you can define addition from the successor operation by induction, and then you can define a to be less than b if there is a positive integer n such that a + n = b. My understanding is that you cannot define addition this way in first-order Peano arithmetic. Instead it is necessary to explicitly talk about addition in the axioms. From here one could also go on to explicitly talk about the order relation in the axioms.
I read that “any” as “for at least one”, rather than as “for every”.
Probably it’s because of the “no group” before it; cf “I can do anything” and “I can’t do anything”. Negations and quantifiers in English sometimes interact in weird ways, making it non-trivial to get the semantics from the syntax.
Wiktionary gives the meanings “at least one” and “no matter what kind”. The first likely doesn’t apply here, as it’s not used in a negation or question. To interpret “no matter what kind” to mean “every” seems like a stretch to me. I really do think the meaning of “any” is ambiguous here. “any” just specifies that we don’t have any further constraints on x. You could replace it with “every” or “at least one”, but not with “every even” or “at least one even”, as that would introduce a new constraint.
I don’t see yet how this connects to the other posts from the epistemology sequence, but it’s definitely nice. I’ve wanted to learn more mathematical logic for some time. I didn’t quite understand why exactly using an axiom schema isn’t as good as using second order logic, before I read this post.
I read that “any” as “for at least one”, rather than as “for every”. That confused me quite a bit. Maybe native speakers won’t have a problem with that, but to me the connection between “any” and “some” is too close.
It’s also not clear to my where the order relation comes from.
I think the point of this post is to demonstrate that logical pinpointing is hard. You might think that the first-order Peano arithmetic axioms logically pinpoint the natural numbers, and what this discussion will end up showing is that they just don’t because of general properties of first-order logic (specifically the Löwenheim–Skolem theorem).
If logically pinpointing something as seemingly simple as the natural numbers depends on something as seemingly nontrivial as understanding the distinction between first-order and second-order logic, then (or so I imagine the argument will continue) we shouldn’t expect logically pinpointing something like morality to be any easier. In fact we have every reason to expect it to be substantially harder.
The definition of the order relation is nontrivial. In second-order Peano arithmetic you can define addition from the successor operation by induction, and then you can define a to be less than b if there is a positive integer n such that a + n = b. My understanding is that you cannot define addition this way in first-order Peano arithmetic. Instead it is necessary to explicitly talk about addition in the axioms. From here one could also go on to explicitly talk about the order relation in the axioms.
Probably it’s because of the “no group” before it; cf “I can do anything” and “I can’t do anything”. Negations and quantifiers in English sometimes interact in weird ways, making it non-trivial to get the semantics from the syntax.
Wiktionary gives the meanings “at least one” and “no matter what kind”. The first likely doesn’t apply here, as it’s not used in a negation or question. To interpret “no matter what kind” to mean “every” seems like a stretch to me. I really do think the meaning of “any” is ambiguous here. “any” just specifies that we don’t have any further constraints on x. You could replace it with “every” or “at least one”, but not with “every even” or “at least one even”, as that would introduce a new constraint.
It doesn’t, but I was hypothesizing that the reason why on the first read it sounded to you as though it did was the negation (“no group”) before it.